Incommensurate charge ordered states in the $\mathit{t-t^{\prime}-J}$ model

TL;DR

This study investigates incommensurate charge ordered states, specifically nodal pair density waves, in the t-t'-J model using Gutzwiller mean field theory, and compares theoretical predictions with STM experimental data on cuprate superconductors.

Contribution

It introduces a detailed analysis of nPDW states in the t-t'-J model and demonstrates their consistency with experimental STM observations, highlighting their potential role in cuprate charge order.

Findings

nPDW states match STM data on form factors and phase differences

Commensurate states do not agree with experimental observations

nPDW states are strong candidates for charge density waves in cuprates

Abstract

We study the incommensurate charge ordered states in the $\mathit{t-t^{\prime}-J}$ model using the Gutzwiller mean field theory on large systems. In particular, we explore the properties of incommensurate charge modulated states referred to as nodal pair density waves (nPDW) in the literature. nPDW states intertwine site and bond charge order with modulated $d$-wave pair order, and are characterized by a nonzero amplitude of uniform pairing; they also manifest a dominant intra-unit cell $d$-density wave form factor. To compare with a recent scanning tunneling microscopy (STM) study [Hamidian \textit{et al.}, Nat. Phys. 3519 (2015)] of the cuprate superconductor BSCCO-2212, we compute the continuum local density of states (LDOS) at a typical STM tip height using the Wannier function based approach. By Fourier transforming Cu and O sub-lattice LDOS we also obtain bias-dependent intra-unit cell form factors and spatial phase difference. We find that in the nPDW state the behavior of form factors and spatial phase difference as a function of energy agrees remarkably well with the experiment. This is in contrast to commensurate charge modulated states, which we show do not agree with experiment. We propose that the nPDW states are good candidates for the charge density wave phase observed in the superconducting state of underdoped cuprates.